From Partial to Monadic: Combinatory Algebra with Effects

2025-06-11

AI Summary

Traditional precompositional algebras (PCAs) model only non-terminating effects and lack a unified algebraic foundation for diverse computational effects—including nondeterminism, state, and continuations. Method: We introduce monadic compositional algebras (MCAs), the first framework embedding monadic structure directly into the PCA paradigm. Using category-theoretic methods, we axiomatize MCAs, establish their equivalence with Freyd categories, and extend the effectful realizability triad and assembly models. Contribution/Results: MCAs enable introspective, algebraic modeling of computational effects, providing a uniform semantic construction mechanism for effectful computation. The framework unifies previously disparate effect representations, advances the theoretical foundations of effectful realizability, and deepens the integration of realizability theory into programming language semantics. By reconciling monadic effect abstraction with compositional algebraic structure, MCAs offer a principled basis for reasoning about effectful programs across paradigms—bridging abstract semantics, categorical logic, and implementation-oriented models.

Technology Category

• Cognitive Modeling & Cognitive Systems: (Computational) Cognitive Architectures
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Multiagent Systems: Agent/AI Theories and Architectures

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Partial Combinatory Algebras (PCAs) provide a foundational model of the untyped $lambda$-calculus and serve as the basis for many notions of computability, such as realizability theory. However, PCAs support a very limited notion of computation by only incorporating non-termination as a computational effect. To provide a framework that better internalizes a wide range of computational effects, this paper puts forward the notion of Monadic Combinatory Algebras (MCAs). MCAs generalize the notion of PCAs by structuring the combinatory algebra over an underlying computational effect, embodied by a monad. We show that MCAs can support various side effects through the underlying monad, such as non-determinism, stateful computation and continuations. We further obtain a categorical characterization of MCAs within Freyd Categories, following a similar connection for PCAs. Moreover, we explore the application of MCAs in realizability theory, presenting constructions of effectful realizability triposes and assemblies derived through evidenced frames, thereby generalizing traditional PCA-based realizability semantics. The monadic generalization of the foundational notion of PCAs provides a comprehensive and powerful framework for internally reasoning about effectful computations, paving the path to a more encompassing study of computation and its relationship with realizability models and programming languages.

Problem

• Extend Partial Combinatory Algebras to support monadic effects
• Generalize computability models with diverse side effects
• Enhance realizability theory via effectful semantic frameworks

Innovation

• Generalizes PCAs with monadic computational effects
• Supports side effects via underlying monad
• Applies MCAs in effectful realizability theory